The dependence of \(\tilde{v}\)  in (REF ) on \(l\)  will be treated
in standard way via Lagrange interpolation of \(\tilde{v}\)  on Chebyshev collocation
points (REF ) as discussed in [1]}. A derivative of
\(\tilde{v}\)  with respect to \(l\)  is then approximated via the
derivative of the Lagrange polynomial. This leads to the action of a
matrix on the vector
\(\tilde{v}=(\tilde{v}(l_{0}),\ldots ,\tilde{v}_{N})\)  (again we use the
same symbol for the function \(\tilde{v}\)  and the vector \(\tilde{v}\) ), the well known
Chebyshev differentiation matrices \(D\) , see e.g.,
[2]}, [3]}. This means with
(REF ) that the
derivatives \(\partial _{x}\)  are approximated by
\(\partial _{x}\approx \frac{2}{\pi c}\mbox{diag}\left(\cos ^{2} \frac{l\pi }{2}\right)D,\) 
